It is interesting to revisit the calculations of the effects of line broadening with regard to the detectability of RRLs as considered by van de Hulst. We try to reproduce his calculations below following the suggestions of Sullivan (1982).

The full width of a Gaussian line at half-intensity, Δν_{ T }, due to thermal broadening is given by

$$\Delta \nu_T = \nu_0 \left(4 \ln 2 \frac{2kT}{Mc^2}\right)^{1/2},$$

(C.1)

where ν

_{0} is the rest frequency of the line and

*M* is the mass of the radiating atom or molecule. Working from the van de Hulst (1945) and the Inglis and Teller (1939) papers, we derived the simplified expression for the Stark width, Δν

_{ vdh }, calculated by van de Hulst to be

$$\Delta\nu_{vdh} \approx \nu_0\left(\frac{3\times 10^6}{\nu_0}\right)^{3/5},$$

(C.2)

where we have used the inverted exponent suggested by Sullivan. In this case, the correct, simplified expression for the Stark width would be

$$\Delta\nu_S \approx \nu_0\left(\frac{3\times 10^6}{\nu_0}\right)^{5/3}.$$

(C.3)

The generalized expression for the line-to-continuum ratio used by van de Hulst to estimate the detectability of RRLs was

$$\frac{I_L}{I_C} = \frac{\nu_0}{\Delta\nu}\frac{h\nu_0}{kT}g,$$

(C.4)

where

*g* is the ratio of the appropriate Gaunt factors and was taken to be ≈ 0.1. Substituting the above expressions for Δν and plotting the results gives Fig.

C.1.

Inspection shows that the *I*_{ L }/*I*_{ C } ratio is indeed low for the calculation of Stark broadening using the 3/5 exponent, in fact, lower than the thermal case by at least two orders of magnitude in the radio wavelength regime from, say, 10^{7} to 10^{11} Hz. Furthermore, the VdH values of *I*_{ L }/*I*_{ C } are so low to constitute unrealistic detection prospects for equipment available in 1945 when the paper appeared. From the presumed calculations shown here, we can easily understand why van de Hulst rejected the possibility of detecting RRLs.

On the other hand, his approximate calculations make sense if we “correct” them by inverting the exponent to 5/3. In these calculations, Fig. C.1 shows that thermal broadening dominates the line widths at frequencies above 900 MHz. Actual observations of the H220α and H109α lines from the Orion nebula fall in the appropriate positions. The former falls below the thermal line and the latter falls on that line. Stark broadening diminishes the *I*_{ L }/*I*_{ C } ratio for the H220α line but has virtually no effect on the H109α line.

We conclude that Sullivan's claim may be correct. An accidental inversion of an exponent would have changed van de Hulst's conclusion of whether or not RRLs would be detectable in radio astronomy.